instruction_seed stringlengths 120 1.22k | source stringclasses 1
value | embeddings listlengths 1.02k 1.02k | mean_positive_score float64 0.78 0.86 | mean_negative_score float64 0.76 0.83 | difference_score float64 0.02 0.04 |
|---|---|---|---|---|---|
Consider a knapsack problem where you have \( n \) items, each with an initial value array \( [x_{10}, x_{20}, x_{30}, \ldots, x_{n0}] \) when the knapsack is empty. However, when you select item \( i \) as the first item, the values change to \( [x_{1i}, x_{2i}, x_{3i}, \ldots, x_{ni}] \), and these values continue to... | automath | [
0.016887709498405457,
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0.015694769099354744,
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-0.0... | 0.808992 | 0.785134 | 0.023858 |
Consider the abelian group \( G = \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_5 \). Let \( g_1 = (1, 1, 0) \) and \( g_2 = (0, 1, 1) \). Determine the order of the subgroup generated by \( g_1 \) and \( g_2 \), and verify if \( G \) is isomorphic to the direct product \( \langle g_1 \rangle \times \langle g_2 \r... | automath | [
0.025053784251213074,
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-0.... | 0.82012 | 0.801702 | 0.018418 |
Consider a category \( C \) defined by the following properties: for any two objects \( X, Y \in C \), there exists an object \( Z \) along with morphisms \( f: X \to Z \) and \( g: Y \to Z \). Define a relation on this category: we say \( C \) is filtered if for any pair of objects \( X \) and \( Y \), such morphisms ... | automath | [
0.024621471762657166,
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0.03234920650720596,
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Given a sequence of real polynomials \( f_n \) in \( \mathbb{R}[x_1, \ldots, x_n]_d \) that are sums of squares, is the limit \( \lim_{n \to \infty} f_n \) guaranteed to be a sum of squares as well? If so, can you provide a constructive method to express this limit as a sum of squares based on the individual expression... | automath | [
0.0025015983264893293,
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Can you provide an example of a pair of chain complexes \( C_* \) and \( D_* \), along with a morphism \( f_*: C_* \to D_* \), such that \( f_n: C_n \to D_n \) is injective for all \( n \in \mathbb{Z} \) but the induced map on homology \( f_*: H_n(C_*) \to H_n(D_*) \) is not injective for some \( n \in \mathbb{Z} \)? | automath | [
0.045691851526498795,
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0.... | 0.804819 | 0.788547 | 0.016271 |
Let \( f_i : \mathbb{R}^n \to \mathbb{R} \) be a sequence of real analytic functions such that for every multi-index \(\alpha\), the partial derivatives \( f_i^\alpha \) converge uniformly to \( f^\alpha \) for some function \( f : \mathbb{R}^n \to \mathbb{R} \). If this condition holds, what can be concluded about the... | automath | [
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Given a directed set \( I \) and for each \( i \in I \), a groupoid \( \mathcal{X}_i \) with functors \( F_{ij} : \mathcal{X}_i \to \mathcal{X}_j \) for \( i \leq j \), how many distinct colimits \( \mathrm{colim} \mathcal{X}_i \) can you construct from this data, assuming the structure satisfies the conditions of bein... | automath | [
0.000006862308964628028,
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... | 0.825817 | 0.807734 | 0.018082 |
Given a Borel function \( f:\mathbb{R} \to \mathbb{R} \) that is the composition of two other Borel functions \( g,h:\mathbb{R} \to \mathbb{R} \), can you provide an example of a pair of Lebesgue measurable functions \( f,g:\mathbb{R} \to \mathbb{R} \) such that \( f \) is Lebesgue measurable while the composition \( f... | automath | [
0.0035993449855595827,
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Given the infinite decimal expansion of $\sqrt{2} = 1.4142135623730950488\ldots$, define $r_n = 1.x_1 x_2 \ldots x_n$ as the truncation of this expansion after the first $n$ decimal digits.
(a) Consider the sets $A_n = (-\infty, r_n)$ for each $n \in \mathbb{N}$. What is the value of $\bigcup_{n \in \mathbb{N}} A_n$ ... | automath | [
0.015283164568245411,
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0.0018419295083731413,
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... | 0.844851 | 0.825685 | 0.019166 |
Given an endomorphism \( T \) of a vector space \( V \) over a field of characteristic \( p \) (where \( p \) could be zero or a prime) with minimal polynomial \( p(t) \) that has distinct roots \( \lambda_1, \ldots, \lambda_n \), do we still have the direct sum decomposition \( V = N_1 \oplus \cdots \oplus N_n \), whe... | automath | [
0.005621866323053837,
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0.039730504155159,
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0.008041951805353165,
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-0... | 0.817736 | 0.79897 | 0.018766 |
Given \( N \) disks \( D_i = (x_i, y_i, r_i) \) for \( i = 1, 2, \ldots, N \) on a plane, each defined by its center \((x_i, y_i)\) and radius \(r_i\), what is the maximum number of disks that can intersect at a single point? | automath | [
0.0380403995513916,
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0.012672667391598225,
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0.03511771187186241,
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Given a sorted array \(\mathcal{A}\) with \(n\) elements (where \(n\) is even), the array is partitioned into two equal halves \(\mathcal{A}_1\) and \(\mathcal{A}_2\). An arbitrary number of odd-indexed elements from \(\mathcal{A}_1\) are swapped with odd-indexed elements from \(\mathcal{A}_2\), while even-indexed elem... | automath | [
0.031776610761880875,
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Let \( G \) and \( H \) be groups presented by \( \langle a, b \mid r_1, r_2, \ldots, r_m \rangle \) and \( \langle a', b' \mid s_1, s_2, \ldots, s_m \rangle \) respectively, where \( \{r_1, \ldots, r_m\} \) and \( \{s_1, \ldots, s_m\} \) are sets of relations, and both groups have exactly two generators. If \( G \) is... | automath | [
0.03577328845858574,
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0.021225308999419212,
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... | 0.816082 | 0.796791 | 0.019291 |
Given the coordinates of the vertices of a triangle: \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \), how can I determine whether the triangle is right-angled, almost right-angled (where moving one vertex by a distance of 1 can make it right-angled), or neither? Please provide a method or formula for making th... | automath | [
0.017817191779613495,
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0.013719829730689526,
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... | 0.81246 | 0.796001 | 0.01646 |
Given a topological space \(X\) with the property that every point \(x \in X\) is contained in a unique maximal anti-discrete subset \(M(x)\), can you provide a specific example of a topological space \(X\) and determine the maximal anti-discrete subsets for at least three distinct points in \(X\)? Additionally, illust... | automath | [
0.03992549702525139,
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0.02030179835855961,
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-0.... | 0.819436 | 0.791732 | 0.027704 |
Let \( X = (0, 1) \), and consider the collection of closed sets \( K_n = \left[0, \frac{1}{n}\right] \) for each integer \( n \geq 1 \). Does this collection \( (K_n)_{n \in \mathbb{N}} \) have the finite intersection property? If so, what is the intersection of all the sets \( K_n \)? Furthermore, does the collection... | automath | [
0.038847338408231735,
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Let \( \triangle ABC \) have vertices at \( A(1, 2) \), \( B(3, 4) \), and \( C(5, 2) \). If a parabola opens upwards has its focus at \( F(0, \frac{1}{4}) \) and is tangent to the lines containing the sides \( AB \), \( BC \), and \( CA \), determine whether the focus \( F \) lies on the circumcircle of triangle \( AB... | automath | [
0.03638170287013054,
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Under what conditions on a Noetherian local ring $R$ and a finite $R$-module $M$ can we assert that the equality $$\text{Ass}(M) = \text{Ass}(M/U) \cup \text{Ass}(U)$$ holds for every maximal submodule $U$ of $M$? | automath | [
0.03028070740401745,
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0.02588645927608013,
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0.015929987654089928,
0.028513342142105103,
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0.0016... | 0.826646 | 0.806846 | 0.0198 |
What is the maximum value of \( n \) such that every 2-coloring of the edges of the complete graph on the set \{1, 2, ..., n\} contains a monochromatic clique with prescribed vertex order, as shown to be at most double-exponential in \( k \) by Shelah, and characterized by the growth rate conjectured by Alon to be expo... | automath | [
0.010157980024814606,
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0.04241470247507095,
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0.0058... | 0.821132 | 0.802532 | 0.0186 |
Given integers \( D \) (the total number of dishes), \( n \) (the number of guests), and a menu size \( d \), what are the conditions under which it is always possible to select \( d \) dishes such that any dish \( k \) not on the menu is preferred by more than half the guests to at least one of the dishes on the menu?... | automath | [
0.014586796052753925,
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-0.0... | 0.844074 | 0.818161 | 0.025913 |
Given two sets of real numbers \( S_1 = [0, N_1] \) and \( S_2 = [0, N_2] \), where \( N_1 \) and \( N_2 \) are positive integers, what is the probability that a value \( x_1 \) chosen uniformly at random from \( S_1 \) and a value \( x_2 \) chosen uniformly at random from \( S_2 \) satisfy the condition \( x_1 + x_2 <... | automath | [
-0.0029800536576658487,
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0.02099134773015976,
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0.012499678879976273,
0.02936599589884281,
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-0... | 0.829241 | 0.808041 | 0.0212 |
Let \( L/K \) be a finite field extension and \( f(x) \in K[x] \) be a non-linear irreducible polynomial. If \( \gcd(\deg(f), [L:K]) = 1 \) and \( \deg(f) > 1 \), what can be concluded about the existence of roots of \( f(x) \) in \( L \)? | automath | [
0.018409065902233124,
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0.03402794525027275,
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0.016014665365219116,
0.014374981634318829,
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-0.0... | 0.822211 | 0.801935 | 0.020276 |
Let \( S \) be the set defined as \( S = \{ x \in y^+: \neg P(x) \} \), where \( y \) is an ordinal and \( P(x) \) is a property. If \( S \) is non-empty and defined such that \( y^+ \) is the successor of \( y \), how can we demonstrate that the set \( S \) must contain elements that create an infinite descending sequ... | automath | [
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-0.... | 0.809479 | 0.781929 | 0.027549 |
For a finite group \( G \) of order \( n \), let \( N(m, i) \) denote the number of \( (m, i) \)-good subsets of \( G \) for fixed values \( m \) and \( i \). Is there a known relationship between the structure of the group \( G \) and the values of \( N(m, i) \)? Specifically, can two groups \( G \) and \( H \) with e... | automath | [
0.017676301300525665,
0.009125149808824062,
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0.017644932493567467,
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0.004410625901073217,
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-0.0... | 0.808932 | 0.786577 | 0.022354 |
Consider the integral
$$I = \int_\Gamma \frac{3z+1}{z^2-z}\,dz$$
evaluated over two paths:
1. $\Gamma_1$: from $1-\sqrt{3}i$ to $1+\sqrt{3}$ along the arc described by $z = 1+\sqrt{3}e^{it}$ for $-\frac{\pi}{2} \leq t \leq 0$.
2. $\Gamma_2$: from $1+\sqrt{3}$ to $1-\sqrt{3}i$ along the arc described by $z = 1+\sq... | automath | [
0.036423347890377045,
0.03430753946304321,
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0.030769795179367065,
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0.033537741750478745,
0.010736878961324692,
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-... | 0.822095 | 0.797345 | 0.024749 |
Given a square and two points \( A \) and \( B \) that are uniformly and independently distributed within the square, a circle is drawn with the segment joining \( A \) and \( B \) as its diameter. What is the probability that the center of the square lies inside this circle? | automath | [
0.026539545506238937,
0.025304719805717468,
-0.045668940991163254,
-0.026169247925281525,
0.03350183367729187,
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0.02863004058599472,
0.03912826627492905,
0.009273010306060314,
0.015431733801960945,
0.043795350939035416,
-0.014099271968007088,
-0.... | 0.816177 | 0.797181 | 0.018996 |
Given two discrete-time signals \( a[n] \) and \( b[n] \) that are both bandlimited and satisfy the Nyquist sampling theorem, derive an expression for the continuous-time signal \( c(t) \) obtained from the product \( c[n] = a[n] b[n] \) and demonstrate why the expression for \( c(t) \) takes the form of a single summa... | automath | [
-0.010307482443749905,
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0.04467657953500748,
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0.01689339615404606,
0.02881884016096592,
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-0.000... | 0.804489 | 0.784349 | 0.020139 |
Let \( V \) be a vector space over a field \( F \) such that the dimension of \( V \) is 2. Given that \( \{u, v\} \) is a basis for \( V \) where \( u \) and \( v \) are distinct nonzero vectors, and \( a \) and \( b \) are nonzero scalars, determine a basis for \( V \) explicitly by finding the expressions for \( \{u... | automath | [
0.03160051256418228,
0.04162970185279846,
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0.04707937315106392,
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0.0021001158747822046,
0.05214199051260948,
0.04600250720977783,
-0.012202169746160507,
0.011968003585934639,
0.022887682542204857,
-0.027878684923052788,
-0.0... | 0.823412 | 0.801903 | 0.021509 |
Given the binary relations \( R \), \( S \), and \( T \) defined as follows:
- \( R \) is defined on \( P(\mathbb{N}) \) by \( A R B \) if and only if \( |A \cup B| \geq 2 \).
- \( S \) is defined on \( \mathbb{Q} \) by \( x S y \) if and only if \( |x| = |y| \) (where \( |q| \) is the greatest integer less than or eq... | automath | [
0.04630711302161217,
0.02331461012363434,
-0.04932926222681999,
-0.020087338984012604,
0.03303547576069832,
-0.011607496067881584,
-0.024185221642255783,
0.061833079904317856,
0.04072266444563866,
-0.0021870224736630917,
0.018363911658525467,
0.018744172528386116,
-0.02356022037565708,
-0.... | 0.816572 | 0.789794 | 0.026778 |
Given two power series \( \sum_{n=1}^{\infty} a_n x^n \) and \( \sum_{n=1}^{\infty} b_n x^n \) with radii of convergence \( R_1 \) and \( R_2 \) respectively, define the radius of convergence \( R \) for the series \( \sum_{n=1}^{\infty} (a_n + b_n) x^n \). If \( R_1 = R_2 \), can \( R \) be determined? Specifically, p... | automath | [
0.008945580571889877,
0.046048812568187714,
-0.044517986476421356,
-0.03254489228129387,
0.010884272865951061,
-0.005320760887116194,
-0.016277797520160675,
0.060815516859292984,
0.0331866592168808,
-0.012036756612360477,
0.012178938835859299,
-0.005680026486515999,
-0.044869061559438705,
... | 0.824742 | 0.803667 | 0.021075 |
Let \( S = \mathbb{R} \) with the standard topology, and let \( A = \{ \frac{1}{n} : n \in \mathbb{N} \} \). Determine the limit points of the set \( A \). What is the set of all limit points of \( A \)? | automath | [
0.03653468191623688,
0.025302963331341743,
-0.04334857314825058,
-0.03770509362220764,
0.0237962044775486,
-0.03104463964700699,
-0.0070449067279696465,
0.03441162034869194,
0.040974970906972885,
-0.017728732898831367,
-0.009491723030805588,
0.030914627015590668,
-0.03387345001101494,
-0.0... | 0.832567 | 0.809148 | 0.02342 |
Given a binary matrix \( M \) of size \( n \times p \) containing zeros and ones, along with two integers \( N \) and \( k \), what is the maximum number of ones that can be covered by \( N \) squares of side length \( k \), and under what conditions can all the ones be covered if \( N \) is varied? Additionally, state... | automath | [
0.017599988728761673,
0.03270881995558739,
-0.05065292865037918,
-0.03236515820026398,
0.03609270602464676,
-0.0266998503357172,
-0.009426039643585682,
0.0629473552107811,
0.021933823823928833,
0.007733848877251148,
0.009919369593262672,
0.03200063481926918,
-0.024221030995249748,
-0.02065... | 0.832256 | 0.808625 | 0.023631 |
Given vectors \(\mathbf{g} = \begin{bmatrix} g_1 \\ g_2 \\ \vdots \\ g_n \end{bmatrix}\) and \(\mathbf{m} = \begin{bmatrix} m_1 \\ m_2 \\ \vdots \\ m_n \end{bmatrix}\), under what conditions can we find a scalar \(\alpha\) such that the resulting vector
\[
\mathbf{g} + \alpha \mathbf{m}
\]
satisfies the inequality
... | automath | [
0.0194766353815794,
0.029587656259536743,
-0.04462170973420143,
-0.039305415004491806,
0.053986042737960815,
-0.004171198233962059,
-0.023914068937301636,
0.08221149444580078,
0.02450992353260517,
-0.008547980338335037,
-0.009877339005470276,
0.016587967053055763,
-0.036489859223365784,
-0... | 0.823507 | 0.805602 | 0.017905 |
Let \( G \) be a group and \( H \) and \( K \) be subgroups of \( G \) such that \( H \leq K \). If \( g \in G \) is an arbitrary element, determine the number of distinct left cosets of \( H \) in \( G \) that can be contained in a single left coset of \( K \). Specifically, express your answer in terms of the index o... | automath | [
0.02970040962100029,
0.03699005767703056,
-0.028302054852247238,
-0.034995187073946,
0.038616910576820374,
-0.025784891098737717,
-0.011681344360113144,
0.05643618479371071,
0.037367746233940125,
-0.00503732543438673,
0.019077422097325325,
0.01936379075050354,
-0.03241362050175667,
-0.0013... | 0.833693 | 0.816531 | 0.017162 |
In the cyclic quadrilateral \(ABCD\), where \(AB = 4\), \(BC = 5\), \(CD = 6\), and \(DA = 7\), compute the sum \(m+n\) where the perimeter of quadrilateral \(A_1B_1C_1D_1\), formed by the feet of the perpendiculars dropped from points \(A\), \(B\), \(C\), and \(D\) to the diagonals, is given in the form \(\frac{m}{n}\... | automath | [
0.024135785177350044,
0.024053864181041718,
-0.04209841787815094,
-0.042483728379011154,
0.033282335847616196,
-0.012645171955227852,
-0.012092527002096176,
0.044026028364896774,
0.04747435078024864,
-0.00190315255895257,
-0.0053869145922362804,
0.04817190393805504,
-0.03658170998096466,
-... | 0.831812 | 0.81034 | 0.021472 |
Why is the closed unit disk \( D = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 \leq 1\} \) not a polyhedron, given that a polyhedron can be defined as the intersection of finitely many half-planes? Specifically, how does the closed unit disk fail to meet the geometric properties or definitions associated with polyhedra in ... | automath | [
0.008437315002083778,
0.0274029690772295,
-0.02143958956003189,
-0.03631829470396042,
0.02362966723740101,
-0.0030439570546150208,
-0.0339030884206295,
0.05778872221708298,
0.03301897272467613,
0.0004185765574220568,
0.015691706910729408,
0.044793300330638885,
-0.02533724159002304,
-0.0201... | 0.809349 | 0.788592 | 0.020757 |
Given the series of functions
1. $$f_n(x) = \frac{x^n}{1 + x^n},$$
determine the supremum of the uniform convergence over the interval \([0, a]\) for \(a < 1\) and over the interval \([0, 1)\).
2. For the series
$$S(x) = \sum_{n=0}^{\infty} (-1)^{n} x^{n} (1-x),$$
compute the exact form of \(S(x)\) for \(0 \l... | automath | [
0.02190898172557354,
0.04367024824023247,
-0.03918442875146866,
-0.02766113542020321,
0.027195792645215988,
-0.009748897515237331,
-0.03812510892748833,
0.0780385434627533,
0.027369976043701172,
-0.001995736500248313,
0.018009284511208534,
0.02044960856437683,
-0.028571156784892082,
-0.004... | 0.82259 | 0.798361 | 0.024229 |
Given an image of size \( n = 256 \) with two Gaussian "bumps" centered at \( \left(\frac{n}{4}, \frac{n}{4}\right) \) and \( \left(\frac{3n}{4}, \frac{3n}{4}\right) \), compute the value of the geodesic distance from an arbitrary point at the top-left corner of the image \( (1,1) \) to the farthest point within the im... | automath | [
-0.00025531387655064464,
0.028424300253391266,
-0.034372687339782715,
-0.03442225977778435,
0.019433483481407166,
-0.011047576554119587,
-0.04163410887122154,
0.07115353643894196,
0.03689388185739517,
-0.003817548742517829,
0.016133103519678116,
0.01903546415269375,
-0.019211724400520325,
... | 0.820624 | 0.803914 | 0.01671 |
Can you provide an example of a measure space \((\Omega, F)\) and a function \(\mu: F \to [0, \infty]\) that is additive but not \(\sigma\)-additive? Specifically, let \(\Omega = \mathbb{N}\) and \(F\) be the power set of \(\mathbb{N}\). Define \(\mu\) such that \(\mu(A) = 0\) if \(A\) is finite and \(\mu(A) = \infty\)... | automath | [
0.015720637515187263,
0.03788719326257706,
-0.0342794694006443,
-0.04703651741147041,
0.03744371235370636,
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0.0811515748500824,
0.03496028482913971,
-0.02326335571706295,
0.007221234031021595,
0.00072933902265504,
-0.047035034745931625,
-0.025766... | 0.814814 | 0.796665 | 0.018148 |
For a finite dimensional algebra \( A \) over an algebraically closed field \( K \), if \( A \) is Morita equivalent to a finite dimensional algebra \( B \) such that \( B/J(B) \) is commutative, can it be shown that the basic algebra of \( B \) is unique up to isomorphism? Additionally, are there known examples of non... | automath | [
0.04566124826669693,
0.037802159786224365,
-0.04072991758584976,
-0.0407700315117836,
0.027945727109909058,
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-0.007127639837563038,
0.066471166908741,
0.03425750881433487,
-0.015484282746911049,
0.019941093400120735,
-0.005720142275094986,
-0.03604188188910484,
-0.001... | 0.796404 | 0.779284 | 0.017121 |
Given the following scenario: You are purchasing items from multiple sellers. Each seller has a fixed shipping cost, and you want to minimize your total cost while meeting certain quantity requirements for specific items. Suppose you have three items offered by seller 1 with costs $c_1 = 5$, $c_2 = 7$, and $c_3 = 3$. Y... | automath | [
-0.01755223236978054,
0.03128240257501602,
-0.048595283180475235,
-0.021377574652433395,
0.05411091819405556,
0.004762319382280111,
-0.03488442301750183,
0.03983721137046814,
0.027206718921661377,
-0.016259949654340744,
0.02280421368777752,
0.02372126467525959,
-0.027182651683688164,
0.006... | 0.80779 | 0.790652 | 0.017138 |
Let \( S = \{1, 2, 3, 4, 5\} \) and define a mapping \( f: S \to T \) where \( T = \{a, b, c\} \) such that
\[
f(1) = a, \quad f(2) = a, \quad f(3) = b, \quad f(4) = c, \quad f(5) = c.
\]
Now let \( X = \{1, 3, 5\} \). Compute the set \( \relcomp {\Img f} {f \sqbrk X} \) and \( f \sqbrk {\relcomp S X} \), then deter... | automath | [
0.03834876790642738,
0.03915564343333244,
-0.04741721600294113,
-0.015253815799951553,
0.0480877086520195,
-0.022258281707763672,
0.005112802144140005,
0.06445412337779999,
0.04012560471892357,
0.0009151677950285375,
0.005294873379170895,
0.033025871962308884,
-0.022645937278866768,
-0.015... | 0.833369 | 0.80585 | 0.027519 |
Consider a nonempty closed convex subset \( K \) of a real Banach space \( E \) with a uniformly Gâteaux differentiable norm. Let \( T: K \to K \) be a nonexpansive mapping and let \( F(T) \) denote the set of fixed points of \( T \). If a sequence \( \{x_n\} \) is defined by the recursive relation
\[
x_{n+1} = \alph... | automath | [
0.01606052741408348,
0.04678674787282944,
-0.01471928134560585,
-0.0388757586479187,
0.033157553523778915,
0.0112106679007411,
-0.00006994955765549093,
0.06596674025058746,
0.028278816491365433,
0.001995969330891967,
0.015768425539135933,
0.01081976480782032,
-0.032924436032772064,
0.00182... | 0.819847 | 0.788729 | 0.031118 |
Given that $N = 4p_1 p_2 \cdots p_k + 3$ where $p_1, p_2, \ldots, p_k$ are all primes of the form $4n + 3$, determine if $N$ can have a prime divisor that is also of the form $4n + 3$. If such a prime exists, what are the conditions under which it can be included in the list of primes $p_1, p_2, \ldots, p_k$? | automath | [
0.022313056513667107,
0.017180129885673523,
-0.024201488122344017,
-0.03213075175881386,
0.037766166031360626,
-0.017466308549046516,
-0.003866667626425624,
0.06986106932163239,
0.007977667264640331,
0.0028366073966026306,
0.014297998510301113,
0.03539811819791794,
-0.03398847207427025,
-0... | 0.836564 | 0.817761 | 0.018803 |
Let \( P(x,y) \) be a relation defined by \( P(x,y) \) meaning "person \( x \) loves person \( y \)". Consider the two logical statements:
1. \( \forall x \exists y \, P(x,y) \) - "For every person \( x \), there exists a person \( y \) such that \( x \) loves \( y \)."
2. \( \exists y \forall x \, P(x,y) \) - "There ... | automath | [
0.024713123217225075,
0.023680109530687332,
-0.04247342050075531,
-0.022411329671740532,
0.035627782344818115,
-0.02613956481218338,
-0.02024717442691326,
0.06109459325671196,
0.024256976321339607,
-0.02097892574965954,
-0.0012926495401188731,
0.02362588234245777,
-0.016133049502968788,
-0... | 0.82676 | 0.804266 | 0.022494 |
Consider a binary quadratic optimization problem defined as follows:
\[
\text{minimize } x^T Q x, \quad \text{subject to } Ax \leq b, \quad x \in \{0,1\}^n
\]
where \(Q\) is an \(n \times n\) integer matrix, \(A\) is an \(n \times n\) integer matrix, and \(b\) is an \(n \times 1\) integer vector.
Let \(d(\lambda) =... | automath | [
0.0030565529596060514,
0.02741122618317604,
-0.04438174143433571,
-0.028435474261641502,
0.027223506942391396,
-0.001345208496786654,
-0.024773595854640007,
0.07090583443641663,
0.03898874670267105,
-0.009010249748826027,
0.0030409684404730797,
0.026587245985865593,
-0.03377117961645126,
-... | 0.82078 | 0.795822 | 0.024958 |
Given a set \( A \) and a \( \star \)-strong complement operator defined on its power set \( \mathcal{P}(A) \), let \( A \) be expressed as a countable union of infinite disjoint subsets. How many such subsets can you form if \( |A| = n \) for some finite \( n \)? Provide the reasoning behind your answer. | automath | [
0.034281790256500244,
0.03347978740930557,
-0.016982879489660263,
-0.022565079852938652,
0.03487372770905495,
-0.020941009745001793,
-0.017777713015675545,
0.06343639642000198,
0.04803596809506416,
-0.0029843072406947613,
0.015622265636920929,
0.0383010171353817,
-0.044339586049318314,
-0.... | 0.833052 | 0.816389 | 0.016663 |
Given a Lie group \( G \) with Lie algebra \( \mathfrak{g} \), consider a Lie subalgebra \( \mathfrak{h} \subseteq \mathfrak{g} \) of dimension \( k \). Using Frobenius' theorem, determine the dimension of the unique maximal integral manifold \( H \) associated with the distribution defined by left translations of \( \... | automath | [
0.026125075295567513,
0.024856556206941605,
-0.040069762617349625,
-0.050359804183244705,
0.04653105139732361,
-0.0004988727159798145,
-0.024683063849806786,
0.04901251196861267,
0.03996942564845085,
-0.006389424204826355,
-0.004566243849694729,
0.024114513769745827,
-0.027690989896655083,
... | 0.814198 | 0.793355 | 0.020843 |
Let \( f: X \to Y \) be a morphism in a category \( \mathcal{C} \). Suppose the kernel \( (K, \phi) \) of \( f \) and the cokernel \( (C, \psi) \) of \( f \) exist in \( \mathcal{C} \). If \( \phi \) is a monomorphism and \( \psi \) is an epimorphism, what can you conclude about the morphism \( f \)? Specifically, is \... | automath | [
0.010619650594890118,
0.0312768779695034,
-0.02200019545853138,
-0.013595621101558208,
0.03647507727146149,
-0.026248307898640633,
0.005619194358587265,
0.09269783645868301,
0.029866809025406837,
-0.01190368551760912,
0.003956958651542664,
0.02466333657503128,
-0.03571319580078125,
-0.0059... | 0.811753 | 0.794181 | 0.017573 |
Consider an infinitely long cylindrical conductor of radius \( a \) carrying a steady current \( I \). Surrounding this conductor is an insulating cylindrical shell of inner radius \( b \) and outer radius \( c \) (where \( b > a \) and \( c > b \)). Using Ampere's Law, find the magnetic field \( B \) at a distance \( ... | automath | [
0.0019289723131805658,
0.04136153310537338,
-0.03687956929206848,
-0.03238441050052643,
0.022576801478862762,
-0.04690786078572273,
-0.053262434899806976,
0.054108746349811554,
0.025879619643092155,
-0.0007931963773444295,
0.007219122257083654,
0.019542820751667023,
-0.03315957635641098,
-... | 0.799743 | 0.782857 | 0.016886 |
Given the Cauchy problem defined by the partial differential equation
\[
\frac{1}{2x}u_x + xu u_y + u^2 = 0
\]
with initial condition
\[
u(x,x) = \frac{1}{x^2} \quad (x > 0),
\]
determine the value of \( u(x,y) \) for points \( (x,y) \) satisfying the relationship
\[
(y + s)^2 = 4 x^2
\]
where \( s = \frac{y ... | automath | [
-0.006372122094035149,
0.04017730802297592,
-0.044372063130140305,
-0.027145419269800186,
0.05425072833895683,
0.004633436445146799,
-0.0068925777450203896,
0.08463817834854126,
0.048587240278720856,
0.010807572863996029,
0.013408130034804344,
0.03422917425632477,
-0.009842075407505035,
-0... | 0.818785 | 0.801087 | 0.017698 |
Given the function \( f(x) = x^3 - 3x + 2 \) defined on the interval \( [-2, 2] \), verify whether the conditions of Rolle's Theorem are satisfied. Specifically, calculate \( f(0) \) and \( f(2) \) and determine if \( f(0) = f(2) \) holds. If it does not, explain why Rolle’s Theorem cannot be applied. If it does, find ... | automath | [
0.044833239167928696,
0.018381809815764427,
-0.0469888374209404,
-0.017534814774990082,
0.025607192888855934,
-0.01334472931921482,
-0.004223926924169064,
0.0919199213385582,
0.01986442320048809,
0.0008686298388056457,
0.01879412680864334,
0.02122637629508972,
-0.03902595490217209,
-0.0281... | 0.819521 | 0.801827 | 0.017694 |
Let \( R \) be an associative unital ring and \( M \) a unitary left \( R \)-module. Is it true that for any \( R \)-module \( M \), the socle \( \text{soc}_{\omega+1}(M) = 0 \) holds when \( \omega \) is the least infinite ordinal? If not, provide a counterexample. Furthermore, what can be said about the statement if ... | automath | [
0.03522901237010956,
0.032971929758787155,
-0.02463194914162159,
-0.0363113135099411,
0.016840491443872452,
-0.023499637842178345,
0.0009859718848019838,
0.0590897835791111,
0.020804863423109055,
-0.014814653433859348,
0.0066902656108140945,
0.010703700594604015,
-0.02136339247226715,
-0.0... | 0.813877 | 0.792856 | 0.02102 |
Is it possible to cover \(\mathbb{R}^n\) with a countable collection of disjoint closed balls of positive radius? Justify your answer. | automath | [
0.008059156127274036,
0.03393705561757088,
-0.025130460038781166,
-0.022717298939824104,
0.024436352774500847,
-0.02371802181005478,
-0.0145233990624547,
0.03622017428278923,
0.0334184430539608,
-0.011698021553456783,
0.010682801716029644,
0.0378427729010582,
-0.043194472789764404,
-0.0188... | 0.836287 | 0.817931 | 0.018356 |
Let \( A \) be a set with a betweenness relation \( B \). Suppose that for any three distinct elements \( a, b, c \in A \), the relation \( (a, b, c) \in B \) implies \( (a, c, b) \in B \) and \( (b, a, c) \in B \). If there are exactly 10 distinct elements in \( A \), how many unique triples \( (x, y, z) \) can be for... | automath | [
0.05591559782624245,
0.02114512212574482,
-0.03823598101735115,
-0.011863015592098236,
0.030313657596707344,
-0.00654030404984951,
-0.02723098360002041,
0.04100145027041435,
0.03280433267354965,
-0.005394227337092161,
0.012761296704411507,
0.03917228430509567,
-0.03539250046014786,
-0.0290... | 0.835822 | 0.814987 | 0.020835 |
Let \( a, b, c \) be non-zero real numbers such that the sum of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) is equal to the sum of the squares of their reciprocals. If the values \( bc^2, ca^2, ab^2 \) are found to be in a certain relationship, which of the following is true?
A) They are in A.P.
B) T... | automath | [
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Let \( R = \mathbb{Z}[\sqrt{10}] \) and \( P = (2, \sqrt{10}) = \{ a + b\sqrt{10} \mid a, b \in \mathbb{Z}, 2 \mid a \} \) be an ideal of \( R \). Determine the structure of the quotient ring \( R/P \). Additionally, is the ideal \( P \) a maximal ideal? | automath | [
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Construct a countable dense subset of the Hilbert space $l^2(\mathbb{N})$ that consists of linearly independent vectors. Provide a specific example of such a subset and demonstrate its properties. | automath | [
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0.00... | 0.830283 | 0.805896 | 0.024387 |
Let \( M \) be a compact manifold without boundary, and let \( f: M \to M \) be a diffeomorphism. Define \( T_f \) as the set of transitive points of \( f \), which is the intersection of the closures of the orbits of all points in \( M \). If \( m \) is the Lebesgue measure on \( M \), under what conditions can we gua... | automath | [
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Given a circle of radius 1, centered at \((x_0, 1)\) moving to the right with an angular speed of \(2\pi\) radians per second and a horizontal speed denoted as \(v\), find the maximum integer \(n\) such that it is possible to position \(n\) equally spaced points on the circumference of the circle without any point inte... | automath | [
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Let \( T \) be a real \( n \times n \) matrix with minimal polynomial \( p_{\mathbb{R}}(t) \) over \( \mathbb{R} \). Define \( p_{\mathbb{C}}(t) \) as the minimal polynomial of \( T \) when considered as a complex matrix over \( \mathbb{C} \). What is the degree of the polynomial \( p_{\mathbb{C}}(t) \) in terms of the... | automath | [
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0.026841256767511368,
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Consider a smooth curve \(\gamma : I \to M\) on a positive definite manifold \(M\) with metric \(g_{ab}\). If the tangent field \(\xi^a\) satisfies \(g_{ab} \xi^a \xi^b = 1\), what is the length of the curve \(\gamma\) parametrized by arc length between two points \(p = \gamma(s_1)\) and \(q = \gamma(s_2)\) on the inte... | automath | [
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0.02352743037045002,
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Given the probabilities \( pw \) (probability of initially picking the winning door) and \( pl \) (probability of initially picking a losing door), along with the conditions on the probabilities of being offered the option to switch ( \( sw \) and \( sl \)), determine the conditions under which the optimal strategy for... | automath | [
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Let \( f(x) = x \cos x \). Determine the signs of the integrals \( \int_{0}^{\frac{\pi}{2}} f(x) \, dx \), \( \int_{\frac{\pi}{2}}^{\pi} f(x) \, dx \), and \( \int_{0}^{\pi} f(x) \, dx \) based on the properties of the function \( f(x) \) over the specified intervals, without evaluating the integrals. What can you conc... | automath | [
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0.01762085035443306,
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Let \( \theta \) be an angle such that \( \sin \theta \neq 0 \). Solve the equation
\[
\sin \theta (2 \cos \theta - 1) = \sin \theta \cos^2 \theta
\]
and determine the complete set of solutions for \( \theta \) in the interval \( [0, 2\pi) \). | automath | [
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0.025317536666989326,
0.01697015017271042,
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-0.01... | 0.828245 | 0.805404 | 0.022841 |
Consider a stochastic process \( (X_t)_{t \in I} \) defined on a measurable space \( (\Omega, \mathcal{A}) \) with values in a Polish space \( E \) equipped with the Borel \( \sigma \)-algebra \( \mathcal{E} \). If we denote by \( \mathbb{P} \) a probability measure on \( (\Omega, \mathcal{A}) \), how does the requirem... | automath | [
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-0.0... | 0.79497 | 0.76995 | 0.025021 |
Let \( f_n(x) = x + \frac{1}{n} \) for \( x \in \mathbb{R} \) and examine the sequences \( g_n(x) = \exp(-(f_n(x))^2) \) and \( h_n(x) = (f_n(x))^2 \).
1. Determine whether the sequence \( g_n \) uniformly converges to the function \( g(x) = \exp(-(f(x))^2) \) as \( n \to \infty \).
2. Determine whether the sequence... | automath | [
0.007757923565804958,
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0.011637029238045216,
0.006960946600884199,
-0.022110672667622566,
0.00... | 0.812292 | 0.790177 | 0.022115 |
Is there any infinitely differentiable function \( f: \mathbb{R} \to \mathbb{R} \) such that \( f'(0) = 1 \) and \( |f^{(n)}(x)| \leq 1 \) for all \( n \in \mathbb{N}_0 \) and \( x \in \mathbb{R} \) other than \( f(x) = \sin(x) \)? If so, provide an example; if not, explain why such a function cannot exist. | automath | [
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0.04537518322467804,
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0.008440081030130386,
0.0047940947115421295,
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Let \( A \) be a finite set with \( n \) distinct points. Define \( d(x, y) = 1 \) if \( x \neq y \) and \( d(x, y) = 0 \) if \( x = y \). Verify whether the topology generated by this metric on \( A \) ensures that every possible finite subset is closed. How many distinct open sets can be formed from this topology? | automath | [
0.03177791088819504,
0.013474653474986553,
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0.01990157924592495,
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0.0035386516246944666,
0.03498297184705734,
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-0... | 0.8357 | 0.800251 | 0.035449 |
In the context of uniquely complemented lattices, if we define a unary operation \(\perp\) by \(\perp(a) = a^\perp\), where \(a^\perp\) is the unique complement of \(a\), and consider the values of the nullary operations \(0\) and \(1\) as the least and greatest elements of the lattice, respectively, can we conclude th... | automath | [
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0.013727388344705105,
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-0.0... | 0.789422 | 0.770867 | 0.018556 |
Given a polynomial \( F(z) = 1 + a_1 z + a_2 z^2 + \cdots + a_m z^m + b_{m+1} z^{m+1} + \cdots + b_n z^n \) whose coefficients satisfy certain conditions, how can you determine the values of \( n \) and the coefficients \( b_{m+1}, b_{m+2}, \ldots, b_n \) such that all zeros of \( F(z) \) lie on a specified star-shaped... | automath | [
0.021983208134770393,
0.04187673330307007,
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0.02856309525668621,
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0.010257842019200325,
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0.0370066799223423,
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-0.... | 0.836208 | 0.809488 | 0.02672 |
Given a circle centered at \( C(C_x, C_y) \) with radius \( r \) and a line defined by two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), derive the conditions under which the line segment \( AB \) intersects the circle. Specifically, what steps must be taken to ensure that only the intersections of the segment with t... | automath | [
0.01945231482386589,
0.02762865647673607,
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0.011652427725493908,
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-0.0... | 0.810344 | 0.782403 | 0.027941 |
Consider two topological spaces $(X, \tau_1)$ and $(X, \tau_2)$ on the same underlying set $X$, where both spaces are separable. If we generate a new topology $\tau$ on $X$ such that the identity maps from $(X, \tau)$ to $(X, \tau_1)$ and $(X, \tau_2)$ are continuous, what can be said about the separability of $(X, \ta... | automath | [
0.022638000547885895,
0.03172256052494049,
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0.010697017423808575,
0.009859643876552582,
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-0.... | 0.814643 | 0.78979 | 0.024853 |
Given an abelian category $\mathcal{A}$ with enough injectives, let $\mathcal{I}^f$ be the strictly full additive subcategory of filtered injective objects. If the functor $K^+(\mathcal{I}^f) \to DF^+(\mathcal{A})$ is an equivalence of triangulated categories, how does the existence of this functor affect the number of... | automath | [
0.030346469953656197,
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0.012118388898670673,
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-0.0... | 0.803392 | 0.785362 | 0.01803 |
Let \( R = k[x,y] \) be the polynomial ring over a field \( k \) and consider the homogeneous ideal \( I = (x^2, xy) \). If we take the element \( a = x \), it is known that the projective dimension \( \operatorname{pd}(R/I) = 2 \) and \( \operatorname{pd}(R/(I+(a))) = 1 \).
Now, let \( a = xy \) instead. Can you det... | automath | [
0.032101791352033615,
0.04432479664683342,
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0.027803173288702965,
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0.02121446654200554,
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-0... | 0.821223 | 0.795236 | 0.025987 |
Is it true that for a compact embedded n-manifold \(\mathcal{M}\) in \(\mathbb{R}^m\) with \(m > n\), the inequality \(d_{\mathcal{M}}(x,y) \leq L \cdot d(x,y)\) holds for some constant \(L > 0\) whenever \(d(x,y)\) is sufficiently small? If so, what is the maximum value of \(L\) that can be established in terms of the... | automath | [
0.011381763033568859,
0.037179891020059586,
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0.035545848309993744,
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0.0027062699664384127,
0.05029121786355972,
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-0... | 0.80585 | 0.778129 | 0.027721 |
Given a sequence of subsets \(\mathcal{A}_n\) of \(\mathbb{N}\), where each subset has either infinitely many elements that contain \(1\) or exclude \(1\), can you construct a subsequence of \(\mathcal{A}_n\) that converges to the set containing \(1\)? If so, determine the general property of such subsequences regardin... | automath | [
0.031831447035074234,
0.027200093492865562,
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0.027475712820887566,
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0.01534187886863947,
0.011195501312613487,
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0.... | 0.835827 | 0.802774 | 0.033053 |
Let \( G \) be an ordered abelian group with rank 1, where the Archimedean property holds. If \( y \) is a positive element of \( G \) and \( x \in G \) such that \( x \neq 0 \), what can you conclude about the existence of an integer \( n \) such that \( n y > x \)? Specifically, if the order of \( G \) is at least 2,... | automath | [
0.04437020793557167,
0.026270363479852676,
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-0.05590539053082466,
0.015740245580673218,
-0.03721260651946068,
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0.04962538182735443,
0.031494397670030594,
-0.005083823576569557,
0.011228309944272041,
0.030550599098205566,
-0.04928698390722275,
-0.0... | 0.830635 | 0.807801 | 0.022833 |
What is the best constant \( C_n \) for the Hardy-Littlewood maximal inequality in the Euclidean space \( \mathbb{R}^n \), and what does the relationship \( C_n = O(n \log n) \) imply about the growth of this constant as \( n \) increases? Additionally, what is the limiting behavior of \( C_n \) in the case of the \( \... | automath | [
-0.006909621879458427,
0.047995537519454956,
-0.043077047914266586,
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0.021631192415952682,
0.008492141030728817,
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0.0712871178984642,
0.05124044418334961,
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0.006583318114280701,
0.00443201744928956,
-0.025328533723950386,
-0.... | 0.809308 | 0.790503 | 0.018805 |
What can be said about the intersection of two transcendental entire functions \( f(x,y) \) and \( g(x,y) \) defined on \( \mathbb{C}^2 \)? In particular, under what conditions can we assert that the set of common zeros \( \{ (x,y) \in \mathbb{C}^2 : f(x,y) = 0 \text{ and } g(x,y) = 0 \} \) is discrete, and how does th... | automath | [
0.02555260807275772,
0.04088311642408371,
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0.03270561620593071,
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0.05093643441796303,
0.04790106788277626,
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0.017809448763728142,
-0.002075913595035672,
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-0.0... | 0.810752 | 0.782095 | 0.028657 |
Let \( G \) be an abelian group with identity element \( e \), and let \( R \) be a ring. Consider a sequence \( \left( x_k \right) \) in \( G \) with \( m \) elements, and let \( \lambda \) be an element of \( R \). If \( \sum_{k=1}^m x_k = e \), what is the value of \( \lambda \circ \left( \sum_{k=1}^m x_k \right) \)... | automath | [
0.02006576769053936,
0.03194575756788254,
-0.03805726394057274,
-0.041028473526239395,
0.03096545860171318,
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0.07058185338973999,
0.014431808143854141,
-0.0012527192011475563,
-0.0010327023919671774,
0.015809370204806328,
-0.0497652143239975,
-0... | 0.832178 | 0.815272 | 0.016906 |
How can we demonstrate that the probability of a particle arriving at point \( m \) after \( n \) steps in a symmetric random walk, given by
\[
p(m,n) = \left(\frac{1}{2}\right)^n \frac{n!}{\left(\frac{n+m}{2}\right)!\left(\frac{n-m}{2}\right)!}
\]
converges to the Gaussian distribution
\[
\lim_{n \to \infty} p(m,... | automath | [
-0.0011352293659001589,
0.0440189428627491,
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0.017534801736474037,
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0.0470641627907753,
0.029427699744701385,
-0.004864905495196581,
0.007949832826852798,
0.029486892744898796,
-0.005252632312476635,
-0... | 0.818729 | 0.799159 | 0.01957 |
Consider the operator \( T: L^p(0,1) \to L^p(0,1) \) defined by \( T x(t) = x'(t) \), where the domain of \( T \) consists of absolutely continuous functions on \( [0,1] \) whose derivatives are in \( L^p(0,1) \). If \( x_n \to x \) in \( L^p(0,1) \) and \( x_n' \to y \) in \( L^p(0,1) \), under what conditions can we ... | automath | [
0.008398523554205894,
0.04994918033480644,
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0.0539531409740448,
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0.062314026057720184,
0.02875448577105999,
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0.0019478985341265798,
0.02256917580962181,
-0.01477336511015892,
-0.... | 0.806831 | 0.784956 | 0.021875 |
Let \( X \) be the set of all positive integers greater than \( 1 \) with a reduction defined by \( a \to b \) if \( a = bc \) for some \( c \in X \). What is the maximum length of a finite chain \( x_1 \to x_2 \to x_3 \to \cdots \to x_n \) that can be formed under this reduction? | automath | [
0.025202102959156036,
0.01587238349020481,
-0.051263902336359024,
-0.022665292024612427,
0.03247091546654701,
-0.018253883346915245,
-0.020435111597180367,
0.045581184327602386,
0.03717115521430969,
-0.020561333745718002,
0.0035395289305597544,
0.04203091189265251,
-0.04027598723769188,
-0... | 0.840954 | 0.82255 | 0.018404 |
In Munkres' Lemma 54.1, he establishes that for any path \( f : [0, 1] \rightarrow B \) starting at \( b_0 \), there exists a unique lifting \( \tilde{f} : [0, 1] \rightarrow E \) that starts at \( e_0 \). Can you provide an explicit example of a covering map \( p : E \rightarrow B \) (with \( B \) being the circle \( ... | automath | [
0.003794478252530098,
0.04648154601454735,
-0.007145436946302652,
-0.05501599982380867,
0.022094309329986572,
-0.015685848891735077,
0.0061421883292496204,
0.07594332844018936,
0.0300140418112278,
0.010554034262895584,
0.015323711559176445,
0.026224669069051743,
-0.008317279629409313,
-0.0... | 0.812636 | 0.783954 | 0.028682 |
Consider the function \( p(x) = \frac{5 + 2^x}{1 - 2^x} \).
1. Determine the horizontal asymptotes of the function as \( x \to \infty \) and \( x \to -\infty \).
2. Verify your findings by calculating the limits \( \lim_{x \to \infty} p(x) \) and \( \lim_{x \to -\infty} p(x) \).
What are the values of these horizont... | automath | [
0.00015805319708306342,
0.04232050105929375,
-0.01942293718457222,
-0.02566048689186573,
0.02010653354227543,
-0.03683901205658913,
-0.026883311569690704,
0.0766003280878067,
0.030474472790956497,
0.005192739423364401,
-0.009301838465034962,
0.014410929754376411,
-0.024590980261564255,
-0.... | 0.823479 | 0.805024 | 0.018455 |
Consider the sequence of sets \( A_n = \{ (x,y) \in \mathbb{R}^2 : (x-n)^2 + y^2 \leq n^2 \} \) for \( n \in \mathbb{N} \).
What is the limit of the sequence \( A_n \) as \( n \to \infty \)? Provide a complete justification for your answer, including whether the limit includes points along the boundary and discussing... | automath | [
0.0072375694289803505,
0.03196083381772041,
-0.026651151478290558,
-0.026420749723911285,
0.011107360944151878,
-0.008234672248363495,
-0.022675205022096634,
0.05805196613073349,
0.042831990867853165,
-0.028153184801340103,
-0.0032506741117686033,
0.046520233154296875,
-0.045290883630514145,... | 0.833438 | 0.801911 | 0.031527 |
Let \( S \) be a non-empty convex set in \( \mathbb{R}^2 \) and let \( f: S \to \mathbb{R} \) be defined as \( f(x, y) = ax + by \) for some constants \( a, b \in \mathbb{R} \). If \( (x_1, y_1), (x_2, y_2) \in S \) and \( \lambda_1, \lambda_2 \geq 0 \) such that \( \lambda_1 + \lambda_2 = 1 \), what can we conclude ab... | automath | [
-0.0018294715555384755,
0.04524919390678406,
-0.059032391756772995,
-0.03352036699652672,
0.044370658695697784,
0.0056750415824353695,
-0.02684214897453785,
0.08231541514396667,
0.03232121095061302,
0.002458949340507388,
0.016371915116906166,
0.02578156441450119,
-0.0446825847029686,
-0.01... | 0.82848 | 0.805548 | 0.022933 |
In triangle \( ABC \) with sides \( AB = 13 \), \( AC = 14 \), and \( BC = 15 \), points \( D \), \( E \), and \( F \) are chosen on sides \( BC \), \( CA \), and \( AB \), respectively, such that \( \frac{AF}{FB} = \frac{2}{5} \) and \( \frac{CE}{EA} = \frac{5}{8} \). Using Ceva's Theorem, calculate the lengths of seg... | automath | [
0.0425124354660511,
0.020121773704886436,
-0.047080982476472855,
-0.01854158565402031,
0.04151758924126625,
-0.03081774339079857,
-0.006536796223372221,
0.0523783415555954,
0.03562936931848526,
0.01790517568588257,
0.014350642450153828,
0.032018307596445084,
-0.03775246813893318,
-0.001254... | 0.828703 | 0.80956 | 0.019143 |
Let \( m \) and \( n \) be positive integers. Consider the function
\[
f(x) = \int_1^x (t-a)^{2n} (t-b)^{2m+1} \, dt
\]
for constants \( a \neq b \). Determine whether \( f(x) \) has a local maximum, local minimum, or neither at the points \( x = a \) and \( x = b \). Provide a justification for your answer using de... | automath | [
0.03451516106724739,
0.039113763719797134,
-0.043261561542749405,
-0.021455736830830574,
0.03582482039928436,
-0.005389132536947727,
-0.028418784961104393,
0.06298365443944931,
0.018415404483675957,
0.010022958740592003,
0.0019753596279770136,
0.043188586831092834,
-0.0430307574570179,
-0.... | 0.835635 | 0.814013 | 0.021622 |
Consider a relation \( R \) defined on a set \( S \) that is reflexive and symmetric, but not transitive. Provide a concrete example of such a relation using real-life scenarios, and justify your example by confirming its properties of reflexivity, symmetry, and non-transitivity. Additionally, offer a brief explanation... | automath | [
0.039825689047575,
0.03254988044500351,
-0.04194265231490135,
-0.03839443624019623,
0.018548106774687767,
-0.018421387299895287,
-0.0025998440105468035,
0.07423905283212662,
0.03816124051809311,
-0.02832734026014805,
0.03850166127085686,
0.028898360207676888,
-0.032420724630355835,
-0.0106... | 0.793748 | 0.776334 | 0.017414 |
Given the method described for approximating the square root of a number, let \( a \) be the extra length needed to approximate \( \sqrt{10} \) using \( 3 + a \). Assuming that the area of the rectangles combined is 1, calculate \( a \) and verify the approximation \( 3 + a \) by checking whether \( (3 + a)^2 \) is gre... | automath | [
0.02711970917880535,
0.047023795545101166,
-0.035514701157808304,
-0.027742600068449974,
0.022969724610447884,
-0.03823236748576164,
-0.0065884068608284,
0.0724572092294693,
0.011532680131494999,
0.005612174980342388,
0.016487115994095802,
0.048495691269636154,
-0.04309907183051109,
-0.020... | 0.830181 | 0.810785 | 0.019396 |
Let \( L : \mathcal{P}_{k} \rightarrow \mathcal{P}_{k+1} \) be defined by \( L p = p''(x) + x p(x) \). For which values of \( k \) does the polynomial \( q(x) = 1 \) belong to the image of \( L \)? Specifically, find the minimum integer \( k \) such that \( q(x) = 1 \notin \text{Im}(L) \). | automath | [
0.014760863035917282,
0.04065001755952835,
-0.03488434478640556,
-0.04246843233704567,
0.037732888013124466,
-0.02165001630783081,
-0.004811721388250589,
0.05686316639184952,
0.030388763174414635,
-0.009747949428856373,
0.006656152196228504,
0.026959257200360298,
-0.019506732001900673,
-0.... | 0.833181 | 0.81477 | 0.018412 |
Let \( f: S \to T \) be a function defined by \( f(x) = 2x \) where \( S = \{1, 2, 3, \ldots, 10\} \). For which values of \( y \in T \) can we be sure that there exists at least one \( x \in S \) such that \( f(x) = y \) if \( T = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\} \)? Additionally, is \( f \) a surjection from \(... | automath | [
0.03080798126757145,
0.05084878206253052,
-0.038010288029909134,
-0.021215878427028656,
0.04452892765402794,
-0.02668781951069832,
-0.026521727442741394,
0.07139203697443008,
0.021776286885142326,
-0.012209598906338215,
0.020911186933517456,
0.028435222804546356,
-0.027917394414544106,
-0.... | 0.83471 | 0.816735 | 0.017975 |
For a sequence of Radon measures \((\mu_n)_n\) defined on a Hausdorff locally compact space \(X\), consider the propositions:
- \((P_1)\): \((\mu_n)_n\) converges vaguely to a Radon measure \(\mu\).
- \((P_2)\): \(\mu(O) \le \underline{\lim} \mu_n(O)\) for all open subsets \(O \subset X\).
- \((P_3)\): \(\mu(K) \ge \o... | automath | [
0.016378525644540787,
0.05399589240550995,
-0.04162939637899399,
-0.03182079270482063,
0.03876505792140961,
-0.008830116130411625,
-0.005472746677696705,
0.06015010178089142,
0.04592185467481613,
0.00021974145784042776,
0.0039995587430894375,
0.014363025315105915,
-0.06291843205690384,
-0.... | 0.811737 | 0.786818 | 0.024919 |
Let \( S \) be the set of squares, defined as \( S = \{ n^2 : n \in \mathbb{N} \} \). According to Szemerédi's theorem, what can be said about the behavior of the quantity \( \alpha_S(N) \), which represents the maximum number of terms of \( S \) contained in arithmetic progressions of length \( N \)? Specifically, wha... | automath | [
0.024707337841391563,
0.025043144822120667,
-0.056015100330114365,
-0.018812745809555054,
0.022473106160759926,
-0.011522497981786728,
-0.025502318516373634,
0.0582093670964241,
0.01807270757853985,
-0.017969708889722824,
-0.003016642527654767,
0.025212595239281654,
-0.04956620931625366,
-... | 0.82451 | 0.798229 | 0.026281 |
Let \( f_n(x) = 1 \) if \( x = n \) and \( 0 \) otherwise, and let \( g_n(x) = 1 \) if \( n \) is even and \( x = 1 \), or \( 1 \) if \( n \) is odd and \( x = 2 \), and \( 0 \) otherwise.
What are the values of \( \lim_{n \to \infty} \sup f_n(x) \) and \( \lim_{n \to \infty} \sup g_n(x) \) for all \( x \)? | automath | [
0.00854706484824419,
0.04034583270549774,
-0.03967917710542679,
-0.03659743815660477,
0.012920601293444633,
-0.007108598947525024,
-0.04376889765262604,
0.06630021333694458,
0.023603569716215134,
-0.0055274381302297115,
-0.0027914405800402164,
0.02674936130642891,
-0.04273401200771332,
-0.... | 0.82947 | 0.812152 | 0.017318 |
Given an even integer \( m \) (where \( 2 \leq m \leq 16 \)) representing the number of distinct points in the plane, and the coordinates of these \( m \) points, determine the maximum number of pairs of parallel line segments that can be formed by coupling the points into pairs. Each point must be paired with exactly ... | automath | [
0.021390313282608986,
0.028125064447522163,
-0.03609851002693176,
-0.029600543901324272,
0.03092005103826523,
0.014311667531728745,
-0.03246116265654564,
0.053996048867702484,
0.029601095244288445,
-0.004402187652885914,
-0.001002085511572659,
0.05127587914466858,
-0.004427865147590637,
-0... | 0.820133 | 0.802469 | 0.017664 |
Let \( w = 7 + 2\sqrt{3} \). Compute the norm \( N(w) \) in \( \mathbb{Z}[\sqrt{3}] \) and determine whether \( w \) is a prime element of \( \mathbb{Z}[\sqrt{3}] \) by examining the irreducibility of \( w \) and the implications of its norm being a prime number in the ordinary integers. If \( w \) is prime, justify yo... | automath | [
0.033220887184143066,
0.025850681588053703,
-0.025987038388848305,
-0.03755659610033035,
0.04413149133324623,
-0.011609293520450592,
-0.001231633359566331,
0.061396729201078415,
0.027483083307743073,
-0.006429092492908239,
0.008902514353394508,
0.030357610434293747,
-0.0471174418926239,
-0... | 0.83293 | 0.815732 | 0.017198 |
Given the equation of a pair of straight lines through the origin represented by
\[
ax^{2} + 2hxy + by^{2} = 0,
\]
the angle \(\theta\) between the lines can be expressed using the formula
\[
\tan \theta = \frac{2\sqrt{h^{2}-ab}}{a+b}.
\]
If the lines are perpendicular, it follows that \(a + b = 0\). The equation... | automath | [
0.007696549873799086,
0.013302989304065704,
-0.05556222051382065,
-0.04224938899278641,
0.04342130571603775,
-0.019524218514561653,
-0.030393818393349648,
0.07086600363254547,
0.03631507605314255,
-0.0029765430372208357,
0.016680873930454254,
0.02151637151837349,
-0.024060692638158798,
-0.... | 0.816179 | 0.797229 | 0.018949 |
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